Help!!! (exponential growth problem)

The earth has 3.2x10^9 acres of arable land available. The world population of 1950 required 10^9 acres to sustain it, and the population of 1980 required 2x10^9 acres. If the required acreage grows at a constant percentage rate, in what year will the population reach the maximum sustainable size?

Question

Help!!! (exponential growth problem)

The earth has 3.2x10^9 acres of arable land available. The world population of 1950 required 10^9 acres to sustain it, and the population of 1980 required 2x10^9 acres. If the required acreage grows at a constant percentage rate, in what year will the population reach the maximum sustainable size?

anonymous · Answer

It takes 30 years for the things to double.

anonymous · Answer

Since we are given a doubling of the area... we'll let 2 be our base. $$
A(t) = C_02^{kt}
$$It takes 30 years to double, so when $t=30$ we want $kt=1$. $$
1 = k(30) \implies k= \frac{1}{30}
$$

anonymous · Answer

Lastly we must solve for $C_0$, given the fact that $A(1950)=10^{9}$: $$ \large
A(1950) = C_0 2^{1950/30} = 10^9
$$

anonymous · Answer

so once we have this how do we determine when the population will reach the max sustainable size

anonymous · Answer

You solve for $t$ when $A(t) = 3.2 \times 10^{9}$

anonymous · Answer

A(t)=3.2×10^9=C02^1950/30
like this???

anonymous · Answer

You still need to solve for $C_0$.

anonymous · Answer

and do i do that by dividing 10^9 by Co^1950/30?
cause if I do that I get a really really small number

anonymous · Answer

......

anonymous · Answer

You do that by solving for it in the equation I already gave.

anonymous · Answer

$$
\large
A(1950) = C_0 \cdot 2^{1950/30} = 10^9
$$

anonymous · Answer

I am not sure how to do that. This is question I have seen. I am sorry

anonymous · Answer

It's called division...

anonymous · Answer

I understand that but what do I divide?

directrix · Answer

http://openstudy.com/users/lands91#/updates/5135a355e4b093a1d94a0af1